Differential forms are used in geometric measure theory to define currents.  One use of currents is as a generalization of submanifolds, with better compactness properties.  That is, it's easier to show that a subsequence of currents converge to a current.  This can give one approach to the minimal surface problem; see [this encyclopediaofmath.org article](http://www.encyclopediaofmath.org/index.php/Geometric_measure_theory#Currents).

Maybe someone else can give some better detail or more uses of currents.